Pharmacy exam 2017-11-27

Degrees: Pharmacy, Biotechnology
Date: November 27, 2017

Question 1

The following diagram show the NO₂ emissions (𝜇g/m³) in Madrid during the month of October, 2017.

The European Standards on Air Quality establish that the average monthly value cannot be over 40 𝜇g/m³ for a healthy environment. Was this requirement met during the month of October? Is the value computed representative of the measurements taken during the month of October?
The Local Government of Madrid has set speed limits on those days with emissions measurements over 72 𝜇g/m³; furthermore, there will be additional parking restrictions if the level is over 92 𝜇g/m³. What percentage of days in October had only speed restrictions?
According to the October sample shown, can we say that the distribution of the NO₂ emissions in the city of Madrid is normally distributed?
Besides the NO₂ level, the Municipal Corporation also checks the level of SO₂, and it has found out that the average level of this substance during October was 2.85 𝜇g/m³, with a standard deviation equa to 0.42 𝜇g/m³. On a day with an NO₂ level of 46, and an SO₂ level of 2.24, which level should be considered higher?
The Air Quality Index (AQI) is computed by multiplying the NO₂ level by 0.95, and adding 30 to the result. What was the average AQI in Madrid during the month of October? Is this value more or less representative than the average NO₂ level?
Are there outliers in the NO₂ emissions in October? Justify your answer.

Use the following data for your computations: $\sum x_i=1945$ 𝜇g/m³,$\sum x_i^2=131575$ (𝜇g/m³)$^2$, $\sum (x_i-\bar x)^3=93995.838$ (𝜇g/m³)³ y $\sum (x_i-\bar x)^4=7766271.021$ (𝜇g/m³)⁴.

Solution

$\bar x=62.7419$ 𝜇g/m³, so the requirement was not met. $s^2=307.8044$ (𝜇g/m³)², $s=17.5444$ 𝜇g/m³, $cv=0.2796$. As the coefficient of variation is less than 0.3 there is a low variability and the mean is quite representative.
$F(72)=0.7097$ and $F(92)=0.9161$, so the percentage of days with only speed restrictions is $20.64%$.
$g_1=0.5615$ and $g_2=-0.3558$. As both of them are between -2 and 2, we can assume that the emissions are normally distributed.
NO₂: $z(46)=-0.9543$. SO₂: $z(2.24)=-1.4524$. Thus, the NO₂ emission is relatively higher.
Let $y=0.95x+30$ the AQI. $\bar y=89.6048$, $s_y=16.6671$, $cv=0.186$. As the coeffitient of variation is lower, the AQI mean is more representative.
$Q_1=49.5816$ 𝜇g/m³, $Q_3=74.0093$ 𝜇g/m³ and $IQR=24.4277$ 𝜇g/m³. Fences: $F_1=12.94$ 𝜇g/m³ and $F_2=110.65$ 𝜇g/m³. Thus, there are outliers.

Question 2

The table below shows the flu incidence rate (per 100,000 people) registered after a number of days from the beginning of the study.

$$ \begin{array}{lrrrrrrrr} \hline \mbox{Days} & 1 & 5 & 8 & 12 & 20 & 26 & 38 & 44\newline \mbox{Flu rate} & 60 & 66 & 71 & 80 & 106 & 132 & 194 & 235\newline \hline \end{array} $$

Estimate the flu incidence rate 50 days after the beginning of the study with a linear regression model.
What is the daily rate of change of the flu incidence rate, according to the linear model computed?
Estimate the incidence rate 50 days after the beginning of the study with an exponential regression model?
Which of the two estimates is more reliable? Why?

Use the following data for your computations ($X=$Days and $Y=$Flu rate): $\sum x_i=154$, $\sum \log(x_i)=19.8494$, $\sum y_j=944$, $\sum \log(y_j)=37.2024$, $\sum x_i^2=4690$, $\sum \log(x_i)^2=60.2309$, $\sum y_j^2=140918$, $\sum \log(y_j)^2=174.8363$, $\sum x_iy_j=25182$, $\sum \log(x_i)y_j=2795.2484$, $\sum x_i\log(y_j)=772.3504$, $\sum \log(x_i)\log(y_j)=96.1974$.

Solution

Linear model of flu rate on days: $\bar x=19.25$ days, $s_x^2=215.6875$ days² . $\bar y=118$ people, $s_y^2=3690.75$ people². $s_{xy}=876.25$ days⋅people. Regression line of flu rate on days: $y=39.7951 + 4.0626x$. $y(50) =242.9247$.
$4.0626$ persons per day.
$\overline{\log(y)}=4.6503$ log(people), $s_{\log(y)}^2=0.2293$ log(people)². $s_{x\log(y)}=7.0255$ days⋅log(people). Exponential model of flu rate on days: $y=e^{4.0233 + 0.0326x}$. $y(50)=284.8357$.
Linear coefficient of determination of flu rate on days $r^2=0.9645$. Exponential coefficient of determination of flu rate on days $r^2=0.9982$. Thus, the exponential model explains a little bit better the evolution of the the flu rate with respect to the number of days.

Last updated on Sep 15, 2020